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arxiv: 1907.08655 · v1 · pith:RHODJFBAnew · submitted 2019-07-19 · 🧮 math.DS · math.NT

Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series

Michel Laurent , Arnaldo Nogueira This is my paper

Pith reviewed 2026-05-24 18:43 UTC · model grok-4.3

classification 🧮 math.DS math.NT
keywords piecewise affine mapsrotation numberHecke-Mahler seriesalgebraic numbersinterval dynamicsrational rotationinjective maps
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The pith

Hecke-Mahler series make the dynamics of 2-interval piecewise affine maps explicit and prove rational rotation numbers for algebraic parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper considers 2-interval piecewise affine increasing maps on [0,1) that are injective but not surjective. These maps are parametrized by three real numbers and possess a rotation number. The authors introduce two functions δ and ϕ, built from Hecke-Mahler series, that fully determine the orbit structure and rotation number of any such map. This explicit description immediately yields the result that algebraic parameters force the rotation number to be rational. A reader would care because the construction converts a dynamical question into an arithmetic one handled by known properties of the series.

Core claim

Let f : [0,1)→[0,1) be a 2-interval piecewise affine increasing map which is injective but not surjective. Such a map f has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of f thanks to two specific functions δ and ϕ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of f is rational, when the three parameters are algebraic numbers.

What carries the argument

The pair of functions δ and ϕ defined via Hecke-Mahler series, which together encode the itinerary and rotation number of the map f for any choice of the three real parameters.

If this is right

  • The rotation number of f equals a value computed directly from δ and ϕ at the three parameters.
  • Algebraic parameters imply that the rotation number is a rational number.
  • The orbit of any point under f is completely determined by the values of δ and ϕ.
  • The construction gives an explicit way to decide whether a given triple of parameters produces periodic or dense orbits.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same series might supply rotation numbers for maps with more than two intervals once suitable extensions of δ and ϕ are defined.
  • Rationality for algebraic parameters suggests that computational checks of rotation numbers could be reduced to algebraic-number arithmetic.
  • The link between Hecke-Mahler series and interval maps may allow transfer of Diophantine approximation results into statements about orbit density.

Load-bearing premise

The complete dynamics of every such map are captured by the two Hecke-Mahler series functions δ and ϕ.

What would settle it

An explicit choice of three algebraic parameters for which the corresponding map f has an irrational rotation number.

Figures

Figures reproduced from arXiv: 1907.08655 by Arnaldo Nogueira, Michel Laurent.

Figure 1
Figure 1. Figure 1: A plot of fλ,µ,δ 2010 Mathematics Subject Classification: 11J91, 37E05. 1 arXiv:1907.08655v1 [math.DS] 19 Jul 2019 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the map ρ 7→ δ(0.9, 0.8, ρ) The series σ(λ, µ, ρ) converges when 0 ≤ ρ < rλ,µ. For fixed λ and µ with 0 < λ < 1, µ > 0, the map ρ 7→ δ(λ, µ, ρ) is increasing in the interval 0 ≤ ρ < rλ,µ and it has a left discontinuity at each rational value (see [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plot of the function φ0.95,0.9,δ,( √ 5−1)/2 in the range 0 ≤ y ≤ 1, where δ = δ(0.95, 0.9,( √ 5 − 1)/2) = 0.6617.... 4 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plot of F1/2,1/2,3/4(x) in the interval −1 ≤ x < 1 It turns out that, for any orbit, this symbolic sequence is either periodic when the rotation number ρ is rational, or a sturmian sequence of slope ρ in the irrational case. We have the following explicit recursion formulae which motivate our definition of the conjugation φ: Lemma 4.1. Let x ∈ R and let (xk)k≥0 be the forward orbit of x by F. For any non￾n… view at source ↗
Figure 5
Figure 5. Figure 5: Case ζ0 > 0 and Case ζ0 = 0. The next proposition provides a partition of the image f n (I), n ≥ 1, into disjoint intervals. It is convenient to consider circular intervals (or circle arcs identifying I with R/Z). For any a, b both belonging to I, we set [a, b) = ( the usual interval [a, b) if a < b [0, b) ∪ [a, 1) if a > b. We write for instance [ζq−1, ζ0) = [0, ζ0) ∪ [ζq−1, 1). Proposition 6. For any int… view at source ↗
Figure 6
Figure 6. Figure 6: Dynamics of the map f with ζ0 > 0 on the left and ζ0 = 0 on the right. The arrows indicate the action of f on the intervals. Proof. Let us consider the partition of I I = [ζ0, ζ1) ∪ [ζ1, ζ2). . . ∪ [ζq−2, ζq−1) ∪ [ζq−1, ζ0) into disjoint circular intervals. The action of f on these intervals is drawn in [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
read the original abstract

Let $f : [0,1)\rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $\delta$ and $\phi$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript studies 2-interval piecewise affine increasing maps f:[0,1)→[0,1) that are injective but not surjective. Such maps admit a rotation number and are parametrized by three real numbers. The authors introduce explicit functions δ and ϕ, defined via Hecke-Mahler series in these parameters, that are asserted to capture the full orbit structure. As an application they conclude that the rotation number is rational whenever the three parameters are algebraic.

Significance. If the constructions and the rationality claim hold, the work would supply an explicit arithmetic criterion linking algebraicity of the parameters to rationality of the rotation number in this class of maps. The explicit use of Hecke-Mahler series to describe the dynamics is a distinctive feature that could open further arithmetic-dynamical investigations.

major comments (1)
  1. [Abstract] Abstract: the central claim that the rotation number is rational for algebraic parameters is asserted without any derivation steps, explicit definitions of δ and ϕ, or supporting arguments; the mathematical support for the claim therefore cannot be evaluated from the available information.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for highlighting the need for clarity in the abstract. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the rotation number is rational for algebraic parameters is asserted without any derivation steps, explicit definitions of δ and ϕ, or supporting arguments; the mathematical support for the claim therefore cannot be evaluated from the available information.

    Authors: Abstracts are concise summaries and are not intended to contain full derivations or explicit function definitions. The manuscript provides the explicit definitions of δ and ϕ via Hecke-Mahler series in the main body, together with the complete arguments establishing that these functions describe the orbit structure and that the rotation number is rational whenever the three parameters are algebraic. The mathematical support for the central claim is therefore contained in the paper and can be evaluated from the full manuscript. revision: no

Circularity Check

0 steps flagged

No circularity; derivation self-contained from series definitions

full rationale

The paper parametrizes the map by three reals, defines δ and ϕ explicitly via Hecke-Mahler series, and derives rationality of the rotation number for algebraic parameters. No quoted step reduces a prediction to a fitted input, self-citation chain, or definitional equivalence; the series supply independent analytic structure outside the target claim. This matches the default non-circular case.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no further details on parameters or assumptions available.

free parameters (1)
  • three real numbers parametrizing the map
    The map f is parametrized by three real numbers as stated in the abstract.
axioms (1)
  • domain assumption Such a map f has a rotation number
    Stated directly in the abstract as a property of the map.

pith-pipeline@v0.9.0 · 5619 in / 1183 out tokens · 28942 ms · 2026-05-24T18:43:20.643157+00:00 · methodology

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